Neat. Surprisingly, there are 388 solutions, and a lot of them look rather unintuitive.
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...Q....
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.....Q..
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Q..B..Q.
Q.......
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..QQB..Q
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Python script: https://gist.github.com/dllu/698d5f71b2b9735c5c462ddf4a2f6fc...
Here's how it works:
0. precompute the attack patterns of each possible queen/bishop location as a bitmask, stored as an integer
1. generate candidate solutions, allowing attack rays to pass through other pieces, by brute forcing the positions of the 5 pieces and taking the bitwise OR of their attacks
2. out of the candidate solutions, check which ones are actually valid taking into account occlusion. Actually, you only need to check if the queen's horizontal attack is blocked by the bishop, as queens cannot block each other (the blocking queen herself has the same attacks so they effectively pass through each other).
More fun facts:
After identifying solutions up to rotation and reflection there are only 49 solutions. No solutions have rotational symmetry, and there is exactly one solution with reflection symmetry (already mentioned by an earlier commenter).
Out of the 49 solution classes, there are 18 distinct queen layouts. The layouts have between 1 and 5 ways to place the bishop to complete the solution. Interestingly, there is exactly one queen layout (up to rotation / reflection) for which there are exactly 2 ways to place the bishop to complete the puzzle.
Even more fun facts: if you change the problem instead to 4 queens and one knight, there is exactly one solution up to symmetries (rotations and flipping). Here it is:
Edit: even more fun facts: if we take the standard piece values of Q=9, R=5, B/N=3, then we can ask for the smallest piece budget that attacks every square. The cheapest possible configuration is 24 points, you can see one with 8 bishops: Which has pleasing symmetry when you view it as a composition of light-square bishops and dark-square bishops.You can't have rotational symmetry with 5 pieces since that would require a piece in the center but the chess board has an even number sized.
I used CP-SAT to enumerate the solutions. A heuristic for "interesting" solutions is those which only admit one valid bishop placement. For example:
Where the bishop lies at the intersection of three queens' horizontal attacks. With these queens, no other bishop placement works.[dead]
I also tried the "place the queens on unique rows and columns".
That got me down to 6 free spaces.
is there a solution where all the pieces are covered as well?
Unfortunately there are none
It's a bit disappointing, then, that the page's action on clicking the "solution" button is just to set the pieces to:
It would be cool if it randomly selected one of those 388, so you could click repeatedly and develop an intuition for what kinds of distributions were a valid solution.ciao, Alex(https://rutar.org/) sent me a super cool visualization of the solutions and I added it.